Find a matrix A with rational entries such that A is not the identity matrix and A^3 = I.
Now I knew I was supposed to use the rational canonical form, but being lazy, the question seemed easy enough to solve by just multiplying everything out. However, I quickly found this is not the case.
What I wish to see is that if anyone can solve the question using a reasoned method. In other words, guess and check is not allowed.
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